In a study conducted by a certain country's Economic Development Board, it was found that the Lorentz curve for the distribution of income of college teachers was described by the function

f(x) =
13/14x^2 + 1\14x
and that of lawyers by the function
g(x) = 5\7x^4 + 2\7x
(a) Compute the coefficient of inequality for each Lorentz curve. (Round your answers to four decimal places.)
college teachers
lawyers

To compute the coefficient of inequality for each Lorentz curve, we need to integrate the given functions and use the resulting area under the curve.

For the college teachers' Lorentz curve:
f(x) = (13/14)x^2 + (1/14)x

To find the coefficient of inequality, we need to calculate the integral of f(x) from 0 to 1 (the range of x values where the curve is defined):
∫[0,1] f(x) dx = ∫[0,1] [(13/14)x^2 + (1/14)x] dx

Integrating f(x) gives us:
∫[0,1] [(13/14)x^2 + (1/14)x] dx = [13/42 * x^3 + 1/28 * x^2] evaluated from 0 to 1

Evaluating the integral gives us:
[13/42 * 1^3 + 1/28 * 1^2] - [13/42 * 0^3 + 1/28 * 0^2]
= 13/42 + 1/28
= 169/588
≈ 0.2874

So, the coefficient of inequality for the college teachers' Lorentz curve is approximately 0.2874.

For the lawyers' Lorentz curve:
g(x) = (5/7)x^4 + (2/7)x

Similarly, we need to calculate the integral of g(x) from 0 to 1:
∫[0,1] g(x) dx = ∫[0,1] [(5/7)x^4 + (2/7)x] dx

Integrating g(x) gives us:
∫[0,1] [(5/7)x^4 + (2/7)x] dx = [5/35 * x^5 + 2/14 * x^2] evaluated from 0 to 1

Evaluating the integral gives us:
[5/35 * 1^5 + 2/14 * 1^2] - [5/35 * 0^5 + 2/14 * 0^2]
= 5/35 + 2/14
= 1/7 + 1/7
= 2/7
≈ 0.2857

So, the coefficient of inequality for the lawyers' Lorentz curve is approximately 0.2857.